Glashow-Salam-Weinberg Electroweak ModelΒΆ

Scope of tutorialΒΆ

In this tutorial, we consider the 1st generation of particles, the fermion-vector boson interactions, the fermion-Higgs boson interactions, the Higgs boson self-interactions, and the vector boson interactions. The quantum chromodynamics (QCD) Lagrangian, which describes the interactions between quarks and gluons, is briefly mentioned at the end.

IntroductionΒΆ

In 1934, Enrico Fermi proposed the following 4-fermion model

GF2(ψ¯eγμψν¯)(ψ¯pγμψn),
to describe Ξ²-decay processes such as nβ†’p+eβˆ’+Ξ½Β―e. The quantity ΟˆΒ―β‰‘Οˆβ€ Ξ³0 is called the Dirac conjugate of the spinor ψ, where the operator † is the Hermitian conjugate, i.e., the transpose and complex conjugate and Ξ³ΞΌ are 4Γ—4 matrices defined by the anti-commutation relations Ξ³ΞΌΞ³Ξ½+Ξ³Ξ½Ξ³ΞΌ=2gΞΌΞ½, where gΞΌΞ½ is the metric tensor of special relativity. GF is called the Fermi constant.

Fermi assumed that β-decay involved vector currents, ψ¯γμψ, but in 1956 T. D. Lee and C. N. Yang noted that no experiment had been performed to test the assumption of parity invariance, that is, the invariance of particle interactions with respect to a reflection in a mirror. They suggested that certain experimental puzzles could be explained if parity was not conserved in the weak interactions. Furthermore, Lee and Yang suggested experiments to test this hypothesis. Shortly thereafter, Chien-Shiung Wu performed the experiment (see https://en.wikipedia.org/wiki/Wu_experiment) and confirmed the violation of parity invariance. The following year, Lee and Yang won the Nobel Prize in Physics. A year later, in 1958, Robert Marshak and his student George Sudarshan developed the vector axial-vector theory (V-A) of the weak interactions, which Richard Feynman and Murray Gell-Mann developed further.

We do not know why the weak interactions violate parity conservation, in particular, why they involve both vector ψ¯γμψ and axial-vector ψ¯γμγ5ψ currents, where Ξ³5≑iΞ³0Ξ³1Ξ³2Ξ³3. Now that we have a better understanding of the weak interactions, however, it would be simpler to say that the weak interactions treat left and right-handed fields differently. But, the V-A jargon has stuck.

The Glashow-Salam-Weinberg (GSW) Langrangian (really a Lagrangian density) is built using the fields below for the 1st generation of leptons and quarks:

L=(Ξ½LeL),eR,Q=(uLdL),uR,dR,where PL=12(1βˆ’Ξ³5),PR=12(1+Ξ³5)are projection operators that split afermion field into its left and right componentsψL,R=PL,Rψ,

which implies ψ=ψL+ψR. Note the absence, in this model, of the field νR.

The GSW model (before addition of the Higgs field) is defined by the Langrangian,

L=LΒ―iΞ³ΞΌ(βˆ‚ΞΌ1+ig1YL2BΞΌ1+ig2Ο„2β‹…WΞΌ)L+QΒ―iΞ³ΞΌ(βˆ‚ΞΌ1+ig1YLQ2BΞΌ1+ig2Ο„2β‹…WΞΌ)Q+eΒ―RiΞ³ΞΌ(βˆ‚ΞΌ+ig1YR2BΞΌ)eR+uΒ―RiΞ³ΞΌ(βˆ‚ΞΌ+ig1YRu2BΞΌ)uR+dΒ―RiΞ³ΞΌ(βˆ‚ΞΌ+ig1YRd2BΞΌ)dRβˆ’14WΞΌΞ½β‹…WΞΌΞ½βˆ’14BΞΌΞ½BΞΌΞ½,

where

1 is a 2Γ—2 unit matrix,Ο„ are the Pauli matrices,the Y are called hypercharges,while the tensors WΞΌΞ½ and BΞΌΞ½ are given byWΞΌΞ½=βˆ‚ΞΌWΞ½βˆ’βˆ‚Ξ½WΞΌβˆ’g2WΞΌΓ—WΞ½,(WΞΌΞ½)i=(βˆ‚ΞΌWΞ½)iβˆ’(βˆ‚Ξ½WΞΌ)iβˆ’g2Ο΅ijkWΞΌjWΞ½k,BΞΌΞ½=βˆ‚ΞΌBΞ½βˆ’βˆ‚Ξ½BΞΌ and WΞΌ=(WΞΌ1,WΞΌ2,WΞΌ3).

The quantities enclosed in the first and second pair of parentheses in the Lagrangian are 2Γ—2 matrices that act on the lepton and quark doublet fields L and Q, respectively, while the parentheses in the remaining fermionic terms enclose 1Γ—1 matrices, that is, scalars, which act on the singlet fields eR, uR, and dR, respectively. However, each field is intrinsically a 4-component Lorentz object: a 4-component spinor in the case of the fermionic fields and a 4-vector in the case of the bosonic fields.

In the GSW Lagrangian, the hypercharges are assigned specific numerical values. Therefore, the Lagrangian has only two free parameters, namely, the couplings g1 and g2.

SU(2)βŠ—U(1) InvarianceΒΆ

The GSW Lagrangian is invariant under the group SU(2)βŠ—U(1). The elements U of SU(2) can be represented as the 2Γ—2 matrices

U=exp⁑(i(Ο„/2)β‹…ΞΈ(x)), whileU=exp⁑(i(Y/2)Ξ±(x)),

represent the elements of U(1). The SU(2) elements are defined by three spacetime-dependent functions, ΞΈ(x)=(ΞΈ1(x), ΞΈ2(x), ΞΈ3(x)), each associated with group generators T1=Ο„1/2, T2=Ο„2/2, and T3=Ο„3/2, respectively, while the U(1) elements are defined by a single function Ξ±(x) and a single generator Y/2.

Invariance under the group SU(2)βŠ—U(1) means that the following replacements leave the Lagrangian unchanged,

SU(2)Lβ†’UL,Qβ†’UQ,eRβ†’eR,uRβ†’uR,dRβ†’dR12Ο„β‹…WΞΌβ†’U(12Ο„β‹…WΞΌ)Uβˆ’1βˆ’1ig2(βˆ‚ΞΌU)Uβˆ’1,BΞΌβ†’BΞΌ,U(1)Lβ†’UL,Qβ†’UQeRβ†’UeR,uRβ†’UuR,dRβ†’UdR,WΞΌβ†’WΞΌ,BΞΌβ†’UBΞΌUβˆ’1βˆ’1ig1(βˆ‚ΞΌU)Uβˆ’1.

It is usually argued that the above local gauge invariance means that the phases of the fields can be set to different values at different events in spacetime and that this makes sense because there is no reason why, for example, Andromedans, ten million years from now, should adopt the same phase convention as Earthlings do now. Personally, I find this argument unconvincing because there are, after all, global symmetries that span the whole of spacetime. For example, the replacements (see for example, https://arxiv.org/pdf/hep-ph/0410370.pdf, p. 9)

Q→eiθ/3Q,uR→eiθ/3uR,dR→eiθ/3dR,andL→eiϕL,eR→eiϕeR,

which correspond to baryon number and lepton number conservation, respectively, are global symmetries that leave the Lagrangian invariant. These global symmetries were not imposed, but appear for free and, in that sense, are accidental. My question is: why is it fine to have global symmetries where certain quantities, such as ΞΈ and Ο• must be the same everywhere and everywhen, that is, Andromedans and Earthlings must use the same values of the phases even though they have never met, and yet it is not fine to have the same gauge field phases everywhere? There is no good reason that I can discern. I think it is better simply to accept the discovery that local guage invariance yields theories, which for reasons not yet clear, are spectacularly successful.

HyperchargesΒΆ

The hypercharge Y, electric charge Q (in units of the proton charge and not to be confused with the quark doublet), and the third component of the isospin I3=Β±1/2 are related by the Gell-Mann Nishijima relation Y=2(Qβˆ’I3). For the left-handed neutrino field, Q=0 and I3=+1/2, therefore, the hypercharge is YL=βˆ’1. For the left-handed electron field, Q=βˆ’1 and I3=βˆ’1/2 and again YL=βˆ’1. On the other hand, since the right-handed electron field is an SU(2) singlet and, therefore, transforms trivially, that is, eRβ†’eR, I3=0 and therefore YR=βˆ’2. We conclude that the hypercharge is a property of the doublet or singlet.

In the quark model, up quarks have electric charge Q=+2/3 (in units of the proton charge), while down quarks have electric charge Q=βˆ’1/3. Therefore, in order to satisfy the Gell-Mann Nishijima relation we must make the assignments YLQ=1/3, YRu=4/3, and YRd=βˆ’2/3.

BosonsΒΆ

If we define XΞΌ=(Ο„/2)β‹…WΞΌ and XΞΌΞ½=(Ο„/2)β‹…WΞΌΞ½, and noting that Ο„aΟ„b=iΟ΅abcΟ„c+Ξ΄ab1 and, therefore, Tr(Ο„aΟ„b)=2Ξ΄ab, we can write

XΞΌΞ½=βˆ‚ΞΌXΞ½βˆ’βˆ‚Ξ½XΞΌ+ig2[XΞΌ,XΞ½],
and, therefore,
WΞΌΞ½β‹…WΞΌΞ½=12Tr(XΞΌΞ½XΞΌΞ½).

In the low energy limit (q2<<MW, where the W boson mass MW∼80GeV), the GSW model reproduces the Fermi model for β-decay provided that we set

GF2=g228MW2.

In [1]:
import sympy as sm
import sympy.functions.special.tensor_functions as t
from sympy.matrices import Matrix
from sympy.physics.matrices import mgamma, msigma
from sympy import I, trigsimp, radsimp

# enable pretty printing of equations
sm.init_printing()

levi = t.LeviCivita
kron = t.KroneckerDelta

Define fieldsΒΆ

Alas, there are differing conventions for defining the fields. In the textbook by Mark Thomson, WΞΌΒ±=12(WΞΌ1βˆ“WΞΌ2), while in the textbook by Gordon Kane (Modern Elementary Particle Physics, 2nd Edition) these fields are defined by WΞΌΒ±=βˆ’12(WΞΌ1Β±WΞΌ2). And in the book by Donnelly, Formaggio, Holstein, Milner, and Surrow (Foundations of Nuclear and Particle Physics), the convention is WΞΌΒ±=12(WΞΌ1Β±WΞΌ2). Here, we'll follow the convention in Thomson's book.

In [2]:
gamma  = sm.Symbol('\gamma^{\mu}', commutative=False)
nubarL = sm.Symbol('\overline{\\nu}_L', commutative=False)
nuL    = sm.Symbol('\\nu_L', commutative=False)
ebarL  = sm.Symbol('\overline{e}_L', commutative=False)
eL     = sm.Symbol('e_L', commutative=False)
ebarR  = sm.Symbol('\overline{e}_R', commutative=False)
eR     = sm.Symbol('e_R', commutative=False)

ubarL  = sm.Symbol('\overline{u}_L', commutative=False)
uL     = sm.Symbol('u_L', commutative=False)
ubarR  = sm.Symbol('\overline{u}_R', commutative=False)
uR     = sm.Symbol('u_R', commutative=False)

dbarL  = sm.Symbol('\overline{d}_L', commutative=False)
dL     = sm.Symbol('d_L', commutative=False)
dbarR  = sm.Symbol('\overline{d}_R', commutative=False)
dR     = sm.Symbol('d_R', commutative=False)

sbarL  = sm.Symbol('\overline{s}_L', commutative=False)
sL     = sm.Symbol('s_L', commutative=False)
sbarR  = sm.Symbol('\overline{s}_R', commutative=False)
sR     = sm.Symbol('s_R', commutative=False)

cbarL  = sm.Symbol('\overline{c}_L', commutative=False)
cL     = sm.Symbol('c_L', commutative=False)
cbarR  = sm.Symbol('\overline{c}_R', commutative=False)
cR     = sm.Symbol('c_R', commutative=False)

B, W1, W2, W3         = sm.symbols("B_\mu, W^1_\mu, W^2_\mu, W^3_\mu")
Bnu, W1nu, W2nu, W3nu = sm.symbols("B_\\nu, W^1_\\nu, W^2_\\nu, W^3_\\nu")
Wplus, Wminus         = sm.symbols("W^+_\mu, W^-_\mu")
Wplusnu,Wminusnu,W3nu = sm.symbols("W^+_\\nu, W^-_\\nu, W^3_\\nu")

Z, A     = sm.symbols('Z_\mu, A_\mu')
Znu, Anu = sm.symbols('Z_\\nu, A_\\nu')

nuL, eL, eR, uL, uR, dL, dR, B, W1, W2, W3
Out[2]:
(Ξ½L, eL, eR, uL, uR, dL, dR, BΞΌ, WΞΌ1, WΞΌ2, WΞΌ3)
In [3]:
Wplus, Wminus, Z, A
Out[3]:
(WΞΌ+, WΞΌβˆ’, ZΞΌ, AΞΌ)

Define couplings, hypercharges, and weak mixing angle ΞΈWΒΆ

In [4]:
g1, g2, thetaW     = sm.symbols('g_1 g_2 \\theta_W')
YQL, YdL, YuR, YdR = sm.symbols('Y^Q_L Y^d_L Y^u_R Y^d_R')
YL, YR, YphiL      = sm.symbols('Y_L Y_R Y_L^\phi')
g1, g2, YL, YR, YQL, YuR, YdR, YphiL, thetaW
Out[4]:
(g1, g2, YL, YR, YLQ, YRu, YRd, YLΟ•, ΞΈW)

Define SU(2) lepton and quark doublets L and QΒΆ

In [5]:
L = Matrix([nuL, eL])
Lbar = Matrix([nubarL, ebarL]).T

Q = Matrix([uL, dL])
Qbar = Matrix([ubarL, dbarL]).T

Lbar, L, Qbar, Q
Out[5]:
([Ξ½Β―LeΒ―L], [Ξ½LeL], [uΒ―LdΒ―L], [uLdL])

Define Ο„β‹…W matrixΒΆ

In [6]:
tau1 = msigma(1)
tau2 = msigma(2)
tau3 = msigma(3)
w    = tau1*W1 + tau2*W2 + tau3*W3
w
Out[6]:
[WΞΌ3WΞΌ1βˆ’iWΞΌ2WΞΌ1+iWΞΌ2βˆ’WΞΌ3]

The fields WΞΌ1βˆ“WΞΌ2 have the form of charged fields, which we write as (WΞΌ1βˆ“WΞΌ2)/2β†’WΞΌΒ±, and are identified as the fields of the charged weak currents.

In [7]:
w = w.subs({W1 - I*W2: sm.sqrt(2)*Wplus,
            W1 + I*W2: sm.sqrt(2)*Wminus})
w
Out[7]:
[WΞΌ32WΞΌ+2WΞΌβˆ’βˆ’WΞΌ3]

Define matrix bΒΆ

In [8]:
b = B*tau1**2 
b
Out[8]:
[BΞΌ00BΞΌ]

Compute matricesΒΆ

ML=g1YL2BΞΌ+g2Ο„2β‹…WΞΌ

and

MQ=g1YLQ2BΞΌ+g2Ο„2β‹…WΞΌ
In [9]:
ML = (g1/2)*YL*b + (g2/2)*w

MQ = (g1/2)*YQL*b + (g2/2)*w

ML, MQ
Out[9]:
([BΞΌYLg12+WΞΌ3g222WΞΌ+g222WΞΌβˆ’g22BΞΌYLg12βˆ’WΞΌ3g22], [BΞΌYLQg12+WΞΌ3g222WΞΌ+g222WΞΌβˆ’g22BΞΌYLQg12βˆ’WΞΌ3g22])

ExpandΒΆ

βˆ’LΒ―Ξ³ΞΌMLLβˆ’QΒ―Ξ³ΞΌMQQ

Note: we had to switch the order of gamma and Lbar for the algebra to look correct.

In [10]:
a = sm.expand(-gamma*Lbar*ML*L)[0] + sm.expand(-gamma*Qbar*MQ*Q)[0]
a
Out[10]:
βˆ’BΞΌYLQg1dΒ―LΞ³ΞΌdL2βˆ’BΞΌYLQg1uΒ―LΞ³ΞΌuL2βˆ’BΞΌYLg1Ξ½Β―LΞ³ΞΌΞ½L2βˆ’BΞΌYLg1eΒ―LΞ³ΞΌeL2βˆ’2WΞΌ+g2Ξ½Β―LΞ³ΞΌeL2βˆ’2WΞΌ+g2uΒ―LΞ³ΞΌdL2βˆ’2WΞΌβˆ’g2dΒ―LΞ³ΞΌuL2βˆ’2WΞΌβˆ’g2eΒ―LΞ³ΞΌΞ½L2βˆ’WΞΌ3g2Ξ½Β―LΞ³ΞΌΞ½L2+WΞΌ3g2dΒ―LΞ³ΞΌdL2+WΞΌ3g2eΒ―LΞ³ΞΌeL2βˆ’WΞΌ3g2uΒ―LΞ³ΞΌuL2

Add the singlet termsΒΆ

βˆ’g12YRBΞΌeΒ―RΞ³ΞΌeRβˆ’g12YRuBΞΌuΒ―RΞ³ΞΌuRβˆ’g12YRdBΞΌdΒ―RΞ³ΞΌdR
In [11]:
l = a
l = l - (g1/2)*YR*B*ebarR*gamma*eR 
l = l - (g1/2)*YuR*ubarR*gamma*B*uR
l = l - (g1/2)*YdR*dbarR*gamma*B*dR
l
Out[11]:
βˆ’BΞΌYLQg1dΒ―LΞ³ΞΌdL2βˆ’BΞΌYLQg1uΒ―LΞ³ΞΌuL2βˆ’BΞΌYRdg1dΒ―RΞ³ΞΌdR2βˆ’BΞΌYRug1uΒ―RΞ³ΞΌuR2βˆ’BΞΌYLg1Ξ½Β―LΞ³ΞΌΞ½L2βˆ’BΞΌYLg1eΒ―LΞ³ΞΌeL2βˆ’BΞΌYRg1eΒ―RΞ³ΞΌeR2βˆ’2WΞΌ+g2Ξ½Β―LΞ³ΞΌeL2βˆ’2WΞΌ+g2uΒ―LΞ³ΞΌdL2βˆ’2WΞΌβˆ’g2dΒ―LΞ³ΞΌuL2βˆ’2WΞΌβˆ’g2eΒ―LΞ³ΞΌΞ½L2βˆ’WΞΌ3g2Ξ½Β―LΞ³ΞΌΞ½L2+WΞΌ3g2dΒ―LΞ³ΞΌdL2+WΞΌ3g2eΒ―LΞ³ΞΌeL2βˆ’WΞΌ3g2uΒ―LΞ³ΞΌuL2

The unification of the electromagnetic and weak nuclear forcesΒΆ

The key idea in the GSW model, as in the original 1961 model by Glashow, is the unification of the electromagnetic and weak interations, which is expressed as follows

(BΞΌWΞΌ3)=(cos⁑θWβˆ’sin⁑θWsin⁑θWcos⁑θW)(AΞΌZΞΌ).

What the unification means concretely is that above a certain critical temperature, the symmetry between the two forces (called the electroweak symmetry) is restored and the two forces merge into a single electroweak force. Below the critical temperature, the electroweak symmetry is said to be spontaneously broken by the change in shape of the potential energy of the Higgs field. While the ensemble of infinitely many vacuum state solutions of the theory respects the electroweak symmetry, the particular vacuum state into which the universe settles does not.

We should be thankful for broken symmetry because without it the universe would be far less diverse and we would not exist.

In [12]:
AZv = Matrix([A, Z])
BWv = Matrix([B, W3])
rot = Matrix([[sm.cos(thetaW),  sm.sin(thetaW)],
             [-sm.sin(thetaW),  sm.cos(thetaW)]])
irot= rot**-1
irot.simplify()
BW  = irot * AZv
AZ  = rot  * BWv
AZ, BW
Out[12]:
([BΞΌcos⁑(ΞΈW)+WΞΌ3sin⁑(ΞΈW)βˆ’BΞΌsin⁑(ΞΈW)+WΞΌ3cos⁑(ΞΈW)], [AΞΌcos⁑(ΞΈW)βˆ’ZΞΌsin⁑(ΞΈW)AΞΌsin⁑(ΞΈW)+ZΞΌcos⁑(ΞΈW)])

Collect terms for AΞΌ and ZΞΌΒΆ

In [13]:
l = l.subs(B,  BW[0])
l = l.subs(W3, BW[1])
l = sm.expand(l)
l = sm.collect(l, A)
l = sm.collect(l, Z)
l
Out[13]:
AΞΌ(βˆ’YLQg1cos⁑(ΞΈW)dΒ―LΞ³ΞΌdL2βˆ’YLQg1cos⁑(ΞΈW)uΒ―LΞ³ΞΌuL2βˆ’YRdg1cos⁑(ΞΈW)dΒ―RΞ³ΞΌdR2βˆ’YRug1cos⁑(ΞΈW)uΒ―RΞ³ΞΌuR2βˆ’YLg1cos⁑(ΞΈW)Ξ½Β―LΞ³ΞΌΞ½L2βˆ’YLg1cos⁑(ΞΈW)eΒ―LΞ³ΞΌeL2βˆ’YRg1cos⁑(ΞΈW)eΒ―RΞ³ΞΌeR2βˆ’g2sin⁑(ΞΈW)Ξ½Β―LΞ³ΞΌΞ½L2+g2sin⁑(ΞΈW)dΒ―LΞ³ΞΌdL2+g2sin⁑(ΞΈW)eΒ―LΞ³ΞΌeL2βˆ’g2sin⁑(ΞΈW)uΒ―LΞ³ΞΌuL2)βˆ’2WΞΌ+g2Ξ½Β―LΞ³ΞΌeL2βˆ’2WΞΌ+g2uΒ―LΞ³ΞΌdL2βˆ’2WΞΌβˆ’g2dΒ―LΞ³ΞΌuL2βˆ’2WΞΌβˆ’g2eΒ―LΞ³ΞΌΞ½L2+ZΞΌ(YLQg1sin⁑(ΞΈW)dΒ―LΞ³ΞΌdL2+YLQg1sin⁑(ΞΈW)uΒ―LΞ³ΞΌuL2+YRdg1sin⁑(ΞΈW)dΒ―RΞ³ΞΌdR2+YRug1sin⁑(ΞΈW)uΒ―RΞ³ΞΌuR2+YLg1sin⁑(ΞΈW)Ξ½Β―LΞ³ΞΌΞ½L2+YLg1sin⁑(ΞΈW)eΒ―LΞ³ΞΌeL2+YRg1sin⁑(ΞΈW)eΒ―RΞ³ΞΌeR2βˆ’g2cos⁑(ΞΈW)Ξ½Β―LΞ³ΞΌΞ½L2+g2cos⁑(ΞΈW)dΒ―LΞ³ΞΌdL2+g2cos⁑(ΞΈW)eΒ―LΞ³ΞΌeL2βˆ’g2cos⁑(ΞΈW)uΒ―LΞ³ΞΌuL2)

In the above expression there is an interaction between the electromagnetic field AΞΌ and the neutrino current Ξ½Β―LΞ³ΞΌΞ½L. However, we have no evidence that such an interaction exists. Therefore, we need to get rid of it. This can be done by setting βˆ’YLg1cos⁑θW = g2sin⁑θW.

In [14]:
l = l.subs(-YL*g1*sm.cos(thetaW), g2*sm.sin(thetaW))
l
Out[14]:
AΞΌ(βˆ’YLQg1cos⁑(ΞΈW)dΒ―LΞ³ΞΌdL2βˆ’YLQg1cos⁑(ΞΈW)uΒ―LΞ³ΞΌuL2βˆ’YRdg1cos⁑(ΞΈW)dΒ―RΞ³ΞΌdR2βˆ’YRug1cos⁑(ΞΈW)uΒ―RΞ³ΞΌuR2βˆ’YRg1cos⁑(ΞΈW)eΒ―RΞ³ΞΌeR2+g2sin⁑(ΞΈW)dΒ―LΞ³ΞΌdL2+g2sin⁑(ΞΈW)eΒ―LΞ³ΞΌeLβˆ’g2sin⁑(ΞΈW)uΒ―LΞ³ΞΌuL2)βˆ’2WΞΌ+g2Ξ½Β―LΞ³ΞΌeL2βˆ’2WΞΌ+g2uΒ―LΞ³ΞΌdL2βˆ’2WΞΌβˆ’g2dΒ―LΞ³ΞΌuL2βˆ’2WΞΌβˆ’g2eΒ―LΞ³ΞΌΞ½L2+ZΞΌ(YLQg1sin⁑(ΞΈW)dΒ―LΞ³ΞΌdL2+YLQg1sin⁑(ΞΈW)uΒ―LΞ³ΞΌuL2+YRdg1sin⁑(ΞΈW)dΒ―RΞ³ΞΌdR2+YRug1sin⁑(ΞΈW)uΒ―RΞ³ΞΌuR2+YLg1sin⁑(ΞΈW)Ξ½Β―LΞ³ΞΌΞ½L2+YLg1sin⁑(ΞΈW)eΒ―LΞ³ΞΌeL2+YRg1sin⁑(ΞΈW)eΒ―RΞ³ΞΌeR2βˆ’g2cos⁑(ΞΈW)Ξ½Β―LΞ³ΞΌΞ½L2+g2cos⁑(ΞΈW)dΒ―LΞ³ΞΌdL2+g2cos⁑(ΞΈW)eΒ―LΞ³ΞΌeL2βˆ’g2cos⁑(ΞΈW)uΒ―LΞ³ΞΌuL2)

So far, so good. Now we wish to recover the known form of the electromagnetic interaction, which for electrons is qAΞΌ(eΒ―LΞ³ΞΌeL+eΒ―RΞ³ΞΌeR). We can do this if we impose the conditions

βˆ’(YR/2)g1cos⁑θW=g2sin⁑θW=q,

where q is the electric charge, or equivalently,

cos⁑θW=βˆ’qg1(YR/2) and sin⁑θW=qg2.

The above expressions imply

q=g1g2(YR/2)g12(YR/2)2+g22,tan⁑θW=βˆ’(YR/2)g1g2
In [15]:
l = l.subs({-YR*g1*sm.cos(thetaW): 2*g2*sm.sin(thetaW) })
l
Out[15]:
AΞΌ(βˆ’YLQg1cos⁑(ΞΈW)dΒ―LΞ³ΞΌdL2βˆ’YLQg1cos⁑(ΞΈW)uΒ―LΞ³ΞΌuL2βˆ’YRdg1cos⁑(ΞΈW)dΒ―RΞ³ΞΌdR2βˆ’YRug1cos⁑(ΞΈW)uΒ―RΞ³ΞΌuR2+g2sin⁑(ΞΈW)dΒ―LΞ³ΞΌdL2+g2sin⁑(ΞΈW)eΒ―LΞ³ΞΌeL+g2sin⁑(ΞΈW)eΒ―RΞ³ΞΌeRβˆ’g2sin⁑(ΞΈW)uΒ―LΞ³ΞΌuL2)βˆ’2WΞΌ+g2Ξ½Β―LΞ³ΞΌeL2βˆ’2WΞΌ+g2uΒ―LΞ³ΞΌdL2βˆ’2WΞΌβˆ’g2dΒ―LΞ³ΞΌuL2βˆ’2WΞΌβˆ’g2eΒ―LΞ³ΞΌΞ½L2+ZΞΌ(YLQg1sin⁑(ΞΈW)dΒ―LΞ³ΞΌdL2+YLQg1sin⁑(ΞΈW)uΒ―LΞ³ΞΌuL2+YRdg1sin⁑(ΞΈW)dΒ―RΞ³ΞΌdR2+YRug1sin⁑(ΞΈW)uΒ―RΞ³ΞΌuR2+YLg1sin⁑(ΞΈW)Ξ½Β―LΞ³ΞΌΞ½L2+YLg1sin⁑(ΞΈW)eΒ―LΞ³ΞΌeL2+YRg1sin⁑(ΞΈW)eΒ―RΞ³ΞΌeR2βˆ’g2cos⁑(ΞΈW)Ξ½Β―LΞ³ΞΌΞ½L2+g2cos⁑(ΞΈW)dΒ―LΞ³ΞΌdL2+g2cos⁑(ΞΈW)eΒ―LΞ³ΞΌeL2βˆ’g2cos⁑(ΞΈW)uΒ―LΞ³ΞΌuL2)

Simplify expressionsΒΆ

We extract the coefficients of the bosonic fields, that is, the currents, and manipulate them into the forms in which they are typically expressed. One of the key manipulations is substituting the appropriate expressions for the hypercharges.

Hypercharge assignmentsΒΆ

Assume the validity of the relation Y=2(Qβˆ’I3), where Q and I3, respectively, are the electric charge in units of the proton charge and the 3rd component of the isospin. Setting Q=0 for the upper component of the lepton doublet, L, which is occupied by the neutrino field, and noting that I3=+1/2 for that component, we find YL=βˆ’1. Repeating this for the lower component, Q=βˆ’1 and I3=βˆ’1/2, we again obtain YL=βˆ’1. And similarly for YR with Q=βˆ’1 and I3=0, we find YR=βˆ’2. Also, YLQ=2(Qfβˆ’I3f)=1/3, YRu=2Qu=4/3, where f is the fermion flavor, and YRd=2Qd=βˆ’2/3.

In [16]:
q = sm.symbols('q')
Qf, Qu, Qd, Qe, Qnu = sm.symbols('Q^f, Q^u, Q^d, Q^e, Q^\\nu')
If3, Id3, Iu3, Ie3, Inu3 = sm.symbols('I^f_3, I^d_3, I^u_3, I^e_3, I^\\nu_3')
q, Qf, Qu, Qd, Qe, Qnu, If3, Id3, Iu3, Ie3, Inu3
Out[16]:
(q, Qf, Qu, Qd, Qe, QΞ½, I3f, I3d, I3u, I3e, I3Ξ½)

Get coefficient of WΞΌ+ΒΆ

In [17]:
Wpluscoeff  = l.coeff(Wplus)
Wpluscoeff  = Wpluscoeff.subs(g2, q/sm.sin(thetaW)).factor(q).simplify()
Wpluscoeff
Out[17]:
βˆ’2q(Ξ½Β―LΞ³ΞΌeL+uΒ―LΞ³ΞΌdL)2sin⁑(ΞΈW)

Get coefficient of WΞΌβˆ’ΒΆ

In [18]:
Wminuscoeff = l.coeff(Wminus)
Wminuscoeff = Wminuscoeff.subs(g2, q/sm.sin(thetaW)).factor(q).simplify()
Wminuscoeff
Out[18]:
βˆ’2q(dΒ―LΞ³ΞΌuL+eΒ―LΞ³ΞΌΞ½L)2sin⁑(ΞΈW)

Get coefficient of AΞΌΒΆ

In [19]:
N, E, U, D = sm.symbols('N, E, U, D')

Acoeff = l.coeff(A).expand()

Acoeff = Acoeff.subs({ebarL*gamma*eL: E,
                      ubarL*gamma*uL: U, 
                      dbarL*gamma*dL: D})
Acoeff = Acoeff.collect(E).collect(U).collect(D)

Acoeff = Acoeff.subs({E:ebarL*gamma*eL, 
                      U: ubarL*gamma*uL, 
                      D: dbarL*gamma*dL})

Acoeff = Acoeff.subs(g1, g2*sm.sin(thetaW)/sm.cos(thetaW))

Acoeff = Acoeff.subs({g2*sm.sin(thetaW): q})

Acoeff
Out[19]:
βˆ’YRdqdΒ―RΞ³ΞΌdR2βˆ’YRuquΒ―RΞ³ΞΌuR2+qeΒ―LΞ³ΞΌeL+qeΒ―RΞ³ΞΌeR+(βˆ’YLQq2βˆ’q2)uΒ―LΞ³ΞΌuL+(βˆ’YLQq2+q2)dΒ―LΞ³ΞΌdL

Implement hypercharge assignments for electromagnetic current.

In [20]:
Acoeff = Acoeff.subs(YuR, 2*Qu)
Acoeff = Acoeff.subs(YdR, 2*Qd)
Acoeff = Acoeff.subs(YQL, 2*(Qf - If3))
Acoeff
Out[20]:
βˆ’QdqdΒ―RΞ³ΞΌdRβˆ’QuquΒ―RΞ³ΞΌuR+qeΒ―LΞ³ΞΌeL+qeΒ―RΞ³ΞΌeR+(βˆ’q(βˆ’2I3f+2Qf)2βˆ’q2)uΒ―LΞ³ΞΌuL+(βˆ’q(βˆ’2I3f+2Qf)2+q2)dΒ―LΞ³ΞΌdL
In [21]:
# up quark current
old = Acoeff.coeff(ubarL*gamma*uL)
new = old.subs({2*If3: 1, Qf: Qu})
Acoeff = Acoeff.subs({old: new})

# down quark current
old = Acoeff.coeff(dbarL*gamma*dL)
new = old.subs({2*If3:-1, Qf: Qd})
Acoeff = Acoeff.subs({old: new}).simplify()

Acoeff
Out[21]:
q(βˆ’QddΒ―LΞ³ΞΌdLβˆ’QddΒ―RΞ³ΞΌdRβˆ’QuuΒ―LΞ³ΞΌuLβˆ’QuuΒ―RΞ³ΞΌuR+eΒ―LΞ³ΞΌeL+eΒ―RΞ³ΞΌeR)

From the above that the electromagnetic current JemΞΌ can be written as

JemΞΌ=βˆ’qβˆ‘f=u,d,eQfψ¯fγμψf,
where ψ=ψL+ψR, noting that PL,RγμPL,R=0.

Get coefficient of ZΞΌΒΆ

Try some algebraic manipulations to simplify expression. Unfortunately, collect does not work on non-commutative symbols. So try the following hack.

In [22]:
Zcoeff = l.coeff(Z)

Zcoeff = Zcoeff.subs({nubarL*gamma*nuL: N, 
                      ebarL*gamma*eL: E,
                      ubarL*gamma*uL: U, 
                      dbarL*gamma*dL: D})
Zcoeff = Zcoeff.collect(E).collect(N).collect(U).collect(D)

Zcoeff = Zcoeff.subs({N:nubarL*gamma*nuL, 
                      E:ebarL*gamma*eL, 
                      U: ubarL*gamma*uL, 
                      D: dbarL*gamma*dL})
Zcoeff = Zcoeff.subs(g1, g2*sm.sin(thetaW)/sm.cos(thetaW))

Zcoeff
Out[22]:
YRdg2sin2⁑(ΞΈW)dΒ―RΞ³ΞΌdR2cos⁑(ΞΈW)+YRug2sin2⁑(ΞΈW)uΒ―RΞ³ΞΌuR2cos⁑(ΞΈW)+YRg2sin2⁑(ΞΈW)eΒ―RΞ³ΞΌeR2cos⁑(ΞΈW)+(YLQg2sin2⁑(ΞΈW)2cos⁑(ΞΈW)βˆ’g2cos⁑(ΞΈW)2)uΒ―LΞ³ΞΌuL+(YLQg2sin2⁑(ΞΈW)2cos⁑(ΞΈW)+g2cos⁑(ΞΈW)2)dΒ―LΞ³ΞΌdL+(YLg2sin2⁑(ΞΈW)2cos⁑(ΞΈW)βˆ’g2cos⁑(ΞΈW)2)Ξ½Β―LΞ³ΞΌΞ½L+(YLg2sin2⁑(ΞΈW)2cos⁑(ΞΈW)+g2cos⁑(ΞΈW)2)eΒ―LΞ³ΞΌeL

Notice that the coefficients of g2cos⁑(ΞΈW) is βˆ’I3. Therefore, let's make those substitutions in the above.

In [23]:
old = Zcoeff.coeff(ubarL*gamma*uL)
new = old.subs({g2*sm.cos(thetaW)/2: g2*Iu3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(dbarL*gamma*dL)
new = old.subs({g2*sm.cos(thetaW)/2: -g2*Id3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(nubarL*gamma*nuL)
new = old.subs({g2*sm.cos(thetaW)/2: g2*Inu3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ebarL*gamma*eL)
new = old.subs({g2*sm.cos(thetaW)/2: -g2*Ie3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

Zcoeff
Out[23]:
YRdg2sin2⁑(ΞΈW)dΒ―RΞ³ΞΌdR2cos⁑(ΞΈW)+YRug2sin2⁑(ΞΈW)uΒ―RΞ³ΞΌuR2cos⁑(ΞΈW)+YRg2sin2⁑(ΞΈW)eΒ―RΞ³ΞΌeR2cos⁑(ΞΈW)+g2(βˆ’2I3Ξ½cos2⁑(ΞΈW)+YLsin2⁑(ΞΈW))Ξ½Β―LΞ³ΞΌΞ½L2cos⁑(ΞΈW)+g2(βˆ’2I3dcos2⁑(ΞΈW)+YLQsin2⁑(ΞΈW))dΒ―LΞ³ΞΌdL2cos⁑(ΞΈW)+g2(βˆ’2I3ecos2⁑(ΞΈW)+YLsin2⁑(ΞΈW))eΒ―LΞ³ΞΌeL2cos⁑(ΞΈW)+g2(βˆ’2I3ucos2⁑(ΞΈW)+YLQsin2⁑(ΞΈW))uΒ―LΞ³ΞΌuL2cos⁑(ΞΈW)

Implement hypercharge assignments for weak neutral current. Note the I3 value for singlet fields is always zero.

Tip: To disable a cell use esc r; use esc y to reactivate it and esc m to go to markdown mode.

In [24]:
old = Zcoeff.coeff(dbarR*gamma*dR)
new = old.subs({YdR: 2*Qd}).cancel().factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ubarR*gamma*uR)
new = old.subs({YuR: 2*Qu}).cancel().factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ebarR*gamma*eR)
new = old.subs({YR: 2*Qe}).cancel().factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(nubarL*gamma*nuL)
new = old.subs({YL: 2*(Qnu - Inu3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(dbarL*gamma*dL)
new = old.subs({YQL: 2*(Qd - Id3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ebarL*gamma*eL)
new = old.subs({YL: 2*(Qe - Ie3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ubarL*gamma*uL)
new = old.subs({YQL: 2*(Qu - Iu3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

Zcoeff
Out[24]:
Qdg2sin2⁑(ΞΈW)dΒ―RΞ³ΞΌdRcos⁑(ΞΈW)+Qeg2sin2⁑(ΞΈW)eΒ―RΞ³ΞΌeRcos⁑(ΞΈW)+Qug2sin2⁑(ΞΈW)uΒ―RΞ³ΞΌuRcos⁑(ΞΈW)βˆ’g2(I3Ξ½βˆ’QΞ½sin2⁑(ΞΈW))Ξ½Β―LΞ³ΞΌΞ½Lcos⁑(ΞΈW)βˆ’g2(I3dβˆ’Qdsin2⁑(ΞΈW))dΒ―LΞ³ΞΌdLcos⁑(ΞΈW)βˆ’g2(I3eβˆ’Qesin2⁑(ΞΈW))eΒ―LΞ³ΞΌeLcos⁑(ΞΈW)βˆ’g2(I3uβˆ’Qusin2⁑(ΞΈW))uΒ―LΞ³ΞΌuLcos⁑(ΞΈW)

The weak neutral current JncΞΌ can be written as

JncΞΌ=2qsin⁑(2ΞΈW)(βˆ‘f=d,e,uQfsin2⁑θWψ¯RfγμψRf+βˆ‘f=Ξ½,d,e,u(βˆ’I3f+Qfsin2⁑θW)ψ¯LfγμψLf).

Final form of GSW lagrangian for 1st generation fermionsΒΆ

Electromagnetic interactionsΒΆ

In [25]:
Acoeff*A
Out[25]:
AΞΌq(βˆ’QddΒ―LΞ³ΞΌdLβˆ’QddΒ―RΞ³ΞΌdRβˆ’QuuΒ―LΞ³ΞΌuLβˆ’QuuΒ―RΞ³ΞΌuR+eΒ―LΞ³ΞΌeL+eΒ―RΞ³ΞΌeR)

Weak charged current interactionsΒΆ

In [26]:
Wpluscoeff*Wplus + Wminuscoeff*Wminus
Out[26]:
βˆ’2WΞΌ+q(Ξ½Β―LΞ³ΞΌeL+uΒ―LΞ³ΞΌdL)2sin⁑(ΞΈW)βˆ’2WΞΌβˆ’q(dΒ―LΞ³ΞΌuL+eΒ―LΞ³ΞΌΞ½L)2sin⁑(ΞΈW)

Weak neutral current interactionsΒΆ

In [27]:
Zcoeff = Zcoeff.factor(g2).subs({g2: q / sm.sin(thetaW)})
Zcoeff*Z
Out[27]:
ZΞΌq(Qdsin2⁑(ΞΈW)dΒ―RΞ³ΞΌdR+Qesin2⁑(ΞΈW)eΒ―RΞ³ΞΌeR+Qusin2⁑(ΞΈW)uΒ―RΞ³ΞΌuR+(βˆ’I3Ξ½+QΞ½sin2⁑(ΞΈW))Ξ½Β―LΞ³ΞΌΞ½L+(βˆ’I3d+Qdsin2⁑(ΞΈW))dΒ―LΞ³ΞΌdL+(βˆ’I3e+Qesin2⁑(ΞΈW))eΒ―LΞ³ΞΌeL+(βˆ’I3u+Qusin2⁑(ΞΈW))uΒ―LΞ³ΞΌuL)sin⁑(ΞΈW)cos⁑(ΞΈW)

The GSW model incorporates quantum electrodynamics (QED), predicts weak charged and neutral current interactions and incorporates the observation that neither the electromagnetic nor the weak neutral currrent interactions change flavor, while the weak charged current interactions always do. And, of course, the model incorporates the parity violation of the weak interactions.

The Higgs SectorΒΆ

The Higgs Lagrangian is given by

LHiggs=(DΞΌΟ•)†(DΞΌΟ•)βˆ’V(ϕ†ϕ),
where
DΞΌ=βˆ‚ΞΌ+ig1YLΟ•2BΞΌ+ig2Ο„2β‹…WΞΌ,V(ϕ†ϕ)=ΞΌ2ϕ†ϕ+Ξ»(ϕ†ϕ)2,and the 4-component Higgs field is given byΟ•=(Ο•1+iΟ•2Ο•3+iΟ•4).

Qualitative description of the Higgs mechanismΒΆ

The stability of the system requires Ξ»>0. In the very early universe, we presume that ΞΌ2>0, in which case ⟨0|Ο•|0⟩=0, that is, the value of the Higgs field in the vacuum is zero. But, at some critical temperature, Tcrit, there is a phase transition in which ΞΌ2 changes sign thereby causing the Higgs potential to develop infinitely many degenerate vacua in each of which the Higgs field is non-zero. Above Tcrit, there is a unique vacuum state and this state respects the symmetry of the Lagrangian. Below Tcrit, the ensemble of vacua respects the symmetry, but, the universe cools into one of the infinitely many vacua, which on its own breaks the symmetry. This is an example of spontaneous symmetry breaking, here electroweak symmetry breaking (EWSB). Spontaneous symmetry breaking is entirely analogous to the breaking of the O(3) symmetry in an (infinite) ferromagnet as it cools below its critical temperature and develops a magnetization in one of infinitely many possible directions, that is, "vacuum" states of the ferromagnet.

The vacuum state of the universe is one in which the Higgs field behaves like a superconductor that attenuates the propagation of the WΞΌ and ZΞΌ fields, thereby rendering these fields massive, but permits the free propagation of the AΞΌ. (The theory is made to do this!) Moreover, the interactions between the Higgs field and the fermions cause a rapid flipping between the left-handed and right-handed fermion fields, which builds up energy within them that we interpret as their mass. The results from the LHC experiment are, so far, consistent with this picture.

Since there is no right-handed neutrino the flipping mechanism cannot occur and therefore the neutrino remains massless in the GSW model. However, neutrinos are now known to have (very small) masses, but, how their masses arise remains an open question.

Quantum numbersΒΆ

We need to specify the value of the hypercharge YLΟ• associated with the doublet Ο•. As above, the hypercharge is chosen so that the Gell-Mann Nishijima relation Y=2(Qβˆ’I3) holds, where the third component of the weak isospin I3=+1/2 for the upper component of a doublet and I3=βˆ’1/2 for the lower component. In the GSW model, it is assumed that the vacuum expectation of Ο• is

⟨0|Ο•|0⟩=12(0v).
Since the vacuum has a net electric charge of zero, we must set Q=0 for the lower component Ο•3+iΟ•4. From YLΟ•=2(Qβˆ’I3) it follows that YLΟ•=+1. This also shows that the upper component, Ο•1+iΟ•2 must be assigned Q=+1, which of course is why it is assumed to be zero in the vacuum otherwise the entire universe would have a huge net positive charge!

Higgs field at low energiesΒΆ

At energies not too far from the vacuum state, we assume that the Higgs field can be approximated as follows

Ο•β‰ˆH=12(0v+h),
where h represents the fluctuations of the Higgs field about its average value vβ‰ˆ246GeV in the vacuum state (the vacuum expectation value). The quanta of the field h are identified with the Higgs bosons discovered at CERN in 2012.

In the following, we shall also need the conjugate field defined by

Hc≑iΟ„2H=12(v+h0).
In [28]:
v = sm.symbols('v')
h = sm.symbols('h')
H = Matrix([0, (v+h)/sm.sqrt(2)])
F = (g1/2)*YphiL*b + (g2/2)*w
Hc=  sm.I*tau2*H
F, H, Hc
Out[28]:
([BΞΌYLΟ•g12+WΞΌ3g222WΞΌ+g222WΞΌβˆ’g22BΞΌYLΟ•g12βˆ’WΞΌ3g22], [02(h+v)2], [2(h+v)20])

NoteΒΆ

The matrix F is Hermitian, that is, F†=F.

In [29]:
y    = F*H
ybar = H.T*F
f = ybar*y
f = f[0]
f = f.subs(YphiL, 1)
f
Out[29]:
WΞΌ+WΞΌβˆ’g22(h+v)24+(h+v)2(BΞΌg12βˆ’WΞΌ3g22)22

Vector boson - Higgs boson interactionsΒΆ

In [30]:
f = f.subs(g1*B/2 - g2*W3/2, Z * sm.sqrt(g1**2 + g2**2)/2)
f
Out[30]:
WΞΌ+WΞΌβˆ’g22(h+v)24+ZΞΌ2(g12+g22)(h+v)28

The terms involving the Higgs boson field are the single and di-Higgs interactions, while the term ∝v2 is the mass term. For a massive vector boson, VΞΌ, the mass term has the form (mV2/2)VΞΌVΞΌ. The term WΞΌ+Wβˆ’ΞΌ is the sum of two terms quadratic in the fields W1 and W2, with the same mass, so the mass term has the form mW2WΞΌ+Wβˆ’ΞΌ. Therefore, in the GSW model the W boson the mass is predicted to be mW=vg2/2, while for the Z boson the prediction is mZ=vg12+g22/2 and, therefore, mW/mZ=g2/g12+g22=cos⁑θW, which modulo higher order corrections agrees with measurements as shown below.

In [31]:
sw2 = 0.2397
sw = sm.sqrt(sw2)
cw = sm.sqrt(1-sw2)
print('cos(theta_W) = %8.3f' % cw)
cos(theta_W) =    0.872
In [32]:
MW=80.385
MZ=91.188
cwp = MW/MZ
print('M_W / M_Z    = %8.3f\t%8.3f' % (cwp, cw / cwp))
M_W / M_Z    =    0.882	   0.989

The Higgs potentialΒΆ

V(H)=ΞΌ2H†Hβˆ’Ξ»(H†H)2

In [33]:
mu, lm = sm.symbols('\mu, \lambda')
V = mu*mu*H.T*H - lm*(H.T*H)**2
V = V.expand()[0]
V
Out[33]:
βˆ’Ξ»h44βˆ’Ξ»h3vβˆ’3Ξ»h2v22βˆ’Ξ»hv3βˆ’Ξ»v44+ΞΌ2h22+ΞΌ2hv+ΞΌ2v22

h-h interactions at low energiesΒΆ

In [34]:
V.subs(mu**2, v**2*lm)
Out[34]:
βˆ’Ξ»h44βˆ’Ξ»h3vβˆ’Ξ»h2v2+Ξ»v44

Yukawa interactions between fermions and the Higgs fieldΒΆ

The interactions between the fermions and the Higgs field are presumed to be Yukawa interactions,

βˆ’2ge(LΒ―HeR+eΒ―RH†L)βˆ’2gu(QΒ―HcuR+uΒ―RHc†Q)βˆ’2gd(QΒ―HdR+dΒ―RH†Q)

Let us pause for a minute to consider the dimensions of the above interactions. In natural units, the Lagrangian has units of [M]4. Since a fermion mass term looks like

mψ¯ψ

and dim(m)=[M], it follows that a fermion field has dimensions [M]3/2. Since the Higgs boson is a scalar field, it has dimensions [M]. Consequently, the Yukawa interactions between fermions and the Higgs boson must be a gauge invariant combination of two fermion fields and the Higgs field. Also, because the Higgs field is a doublet, it is necessary to have it interact with the Dirac conjugate of a doublet field and a singlet field.

Notice also that in order to avoid a flavor changing Yukawa interaction, we need to fudge the interaction of the up quark field with the Higgs field by using the conjugate field Hc rather than H!

In [35]:
ge, me = sm.symbols('g_e, m_e')
gu, mu, gd, md = sm.symbols('g_u, m_u, g_d, m_d')
gu, mu, gd, md
ge, me, gu, mu, gd, md
Out[35]:
(ge, me, gu, mu, gd, md)
In [36]:
Hc = sm.I*tau2*H
Qbar, Q, Hc, Hc.T
Out[36]:
([uΒ―LdΒ―L], [uLdL], [2(h+v)20], [2(h+v)20])
In [43]:
Le = -ge*(Lbar*H*eR + ebarR*H.T*L)[0]*sm.sqrt(2)
Le = Le.expand()
Le = Le.simplify().expand().collect(h*ge).collect(ge*v)
Le = Le.subs(ge*v, me)
Le = Le.subs(ge, me/v)
Le
Out[43]:
hme(βˆ’eΒ―ReLβˆ’eReΒ―L)v+me(βˆ’eΒ―ReLβˆ’eReΒ―L)
In [45]:
Lq = -gd*(Qbar*H*dR  + dbarR*H.T*Q)[0]*sm.sqrt(2) \
     -gu*(Qbar*Hc*uR + ubarR*Hc.T*Q)[0]*sm.sqrt(2)
Lq = Lq.expand()
Lq = Lq.subs(gu*v, mu)
Lq = Lq.subs(gd*v, md)
Lq = Lq.collect(gd*h)
Lq = Lq.collect(gu*h)
Lq = Lq.collect(md)
Lq = Lq.collect(mu)
Lq = Lq.subs(gu, mu/v)
Lq = Lq.subs(gd, md/v)
Lq
Out[45]:
hmd(βˆ’dΒ―RdLβˆ’dRdΒ―L)v+hmu(βˆ’uΒ―RuLβˆ’uRuΒ―L)v+md(βˆ’dΒ―RdLβˆ’dRdΒ―L)+mu(βˆ’uΒ―RuLβˆ’uRuΒ―L)

We see that the Yukawa couplings, together with the particular choice of the value of the Higgs field in the vacuum state, yields two important consequences as alluded to above: mass terms for the fermions and interactions between the Higgs boson and fermions that are proportional to the fermion mass and inversely proportional to the vacuum expectation value v. We conclude that the Higgs mechanism generates mass terms for the weak vector bosons as well as the fermions while preserving guage invariance.

The Cabibbo HypothesisΒΆ

So far we have considered only the 1st generation of particles and we have implicitly assumed that there is no "crosstalk" between the generations. However, there are decays that can be explained if "crosstalk" exists between generations. For example, the existence of decays such as

Ξ›(d,u,s)β†’p(d,u,u)eβˆ’Ξ½Β―e
are readily explained by supposing that the W boson can convert s-quarks to u-quarks. Further clues are provided by the rates of decays such as
Ο€+β†’ΞΌ+Ξ½ΞΌ,K+β†’ΞΌ+Ξ½ΞΌ,
which are measured to be in the ratio
Ξ“(Ο€+β†’ΞΌ+Ξ½ΞΌ)Ξ“(K+β†’ΞΌ+Ξ½ΞΌ)β‰ˆ120.
The above decays can be described with the following Feynman diagrams, decays in which the couplings g2? differ from g2 by the appropriate factor in order to match the measured ratio. These observations and many others were explained by the Italian physicist Cabibbo who suggested that the lower component of the quark doublet, that is, the down quark field, be replaced with the combination
dLβ†’dLβ€²=cos⁑θCdL+sin⁑θCsL,
where ΞΈC is called the Cabibbo angle. With a suitable choice of the Cabibbo angle (about 12o), the Cabibbo hypothesis worked well. However, as is evident from the form of the down component of the left-handed part of weak neutral current,
(I3dβˆ’Qdsin2⁑(ΞΈW))dΒ―LΞ³ΞΌdL,
the Cabibbo hypothesis predicts the existence of flavor-changing neutral currents (FCNC) at a rate in sharp disagreement with observations.

In [55]:
thetaC = sm.symbols('theta_C')
xbarL, xL = sm.symbols('\overline{d^\prime}_L, '\
                       'd^\prime_L', commutative=False)
Nc = xbarL * gamma * xL 
Nc = Nc.subs({xL: sm.cos(thetaC)*dL + sm.sin(thetaC)*sL,
              xbarL: sm.cos(thetaC)*dbarL + sm.sin(thetaC)*sbarL})
Nc = Nc.expand()
Nc = (Id3 - Qd * sm.sin(thetaW)**2) * Nc
Nc
Out[55]:
(I3dβˆ’Qdsin2⁑(ΞΈW))(sin2⁑(ΞΈC)sΒ―LΞ³ΞΌsL+sin⁑(ΞΈC)cos⁑(ΞΈC)dΒ―LΞ³ΞΌsL+sin⁑(ΞΈC)cos⁑(ΞΈC)sΒ―LΞ³ΞΌdL+cos2⁑(ΞΈC)dΒ―LΞ³ΞΌdL)

The GIM MechanismΒΆ

In 1970, Sheldon Glashow, John Iliopoulos and Luciano Maiani (GIM) noted that if the strange quark were the lower component of another doublet,

(cLsL),
whose upper component would be a new quark field, dubbed "charm", and if one generalized Cabibbo's idea and assumed that the lower components of the two quark doublets were mixed as follows
(dLβ€²sLβ€²)=(cos⁑θCsin⁑θCβˆ’sin⁑θCcos⁑θC)(dLsL),
then the replacement
sLβ†’sLβ€²=βˆ’sin⁑θCdL+cos⁑θCsL
would yield exact cancellation of the unwanted FCNC interactions, as demonstrated below.

Implement GIM MechanismΒΆ

In [56]:
ybarL, yL = sm.symbols('\overline{s^\prime}_L, '\
                                  's^\prime_L', 
                                  commutative=False)
# add the GIM terms to the Cabbibo terms
yy = ybarL * gamma * yL
yy = yy.subs({yL:-sm.sin(thetaC)*dL + sm.cos(thetaC)*sL,
              ybarL:-sm.sin(thetaC)*dbarL + sm.cos(thetaC)*sbarL})
yy = yy.expand()
Nc += (Id3 - Qd * sm.sin(thetaW)**2) * yy

# and simplify
Nc = Nc.simplify()
Nc
Out[56]:
(I3dβˆ’Qdsin2⁑(ΞΈW))(dΒ―LΞ³ΞΌdL+sΒ―LΞ³ΞΌsL)

By introducing a 2nd doublet, the GIM mechanism neatly renders the weak neutral currents diagonal again and, moreover, predicts the existence of a 4th quark, the charm quark. This prediction was spectacularly confirmed in 1974 with the discovery of the J/ψ at Brookhaven and SLAC.

Unfortunately, however, no one understands why mixing across the (three) generations should exist and what determines the strength of the mixing.

Vector boson self-interactionsΒΆ

The pure vector boson Lagrangian is given by

LVB=βˆ’14WΞΌΞ½β‹…WΞΌΞ½βˆ’14BΞΌΞ½BΞΌΞ½,
where
(WΞΌΞ½)i=(βˆ‚ΞΌWΞ½βˆ’βˆ‚Ξ½WΞΌ)iβˆ’g2Ο΅ijkWΞΌjWΞ½k,BΞΌΞ½=βˆ‚ΞΌBΞ½βˆ’βˆ‚Ξ½BΞΌ.

From

(BΞΌWΞΌ3)=(cos⁑θWβˆ’sin⁑θWsin⁑θWcos⁑θW)(AΞΌZΞΌ)WΞΌΒ±=12(WΞΌ1βˆ“WΞΌ2)
we obtain
W1=(Wβˆ’+W+)/2W2=(Wβˆ’βˆ’W+)/(2)W3=sin⁑θWA+cos⁑θWZB=cos⁑θWAβˆ’sin⁑θWZ,
which we use below to rewrite the vector boson Lagrangian in terms of the usual fields.

In [68]:
Wplusmu, Wminusmu = sm.symbols("W^+_\mu, W^-_\mu")
Wplusnu, Wminusnu = sm.symbols("W^+_\\nu, W^-_\\nu")

Zmu, Amu          = sm.symbols('Z_\mu, A_\mu')
Znu, Anu          = sm.symbols('Z_\\nu, A_\\nu')

dmuWplusnu        = sm.Symbol('\partial_\mu W^+_\\nu')
dnuWplusmu        = sm.Symbol('\partial_\\nu W^+_\mu')

dmuWminusnu       = sm.Symbol('\partial_\mu W^-_\\nu')
dnuWminusmu       = sm.Symbol('\partial_\\nu W^-_\mu')

dmuZnu            = sm.Symbol('\partial_\mu Z_\\nu')
dnuZmu            = sm.Symbol('\partial_\\nu Z_\mu')

dmuAnu            = sm.Symbol('\partial_\mu A_\\nu')
dnuAmu            = sm.Symbol('\partial_\\nu A_\mu')
In [69]:
Vmu = Matrix([(Wminusmu + Wplusmu)/sm.sqrt(2), 
              (Wminusmu - Wplusmu)/sm.sqrt(2), 
              sm.sin(thetaW)*Amu + sm.cos(thetaW)*Zmu]) 

Vnu = Matrix([(Wminusnu + Wplusnu)/sm.sqrt(2), 
              (Wminusnu - Wplusnu)/sm.sqrt(2),  
              sm.sin(thetaW)*Anu + sm.cos(thetaW)*Znu])
Vmu, Vnu
Out[69]:
([2(WΞΌ++WΞΌβˆ’)22(βˆ’WΞΌ++WΞΌβˆ’)2AΞΌsin⁑(ΞΈW)+ZΞΌcos⁑(ΞΈW)], [2(WΞ½++WΞ½βˆ’)22(βˆ’WΞ½++WΞ½βˆ’)2AΞ½sin⁑(ΞΈW)+ZΞ½cos⁑(ΞΈW)])
In [70]:
dVnu = Matrix([(dnuWminusmu + dnuWplusmu)/sm.sqrt(2), 
               (dnuWminusmu - dnuWplusmu)/sm.sqrt(2), 
               sm.sin(thetaW)*dnuAmu + sm.cos(thetaW)*dnuZmu]) 

dVmu = Matrix([(dmuWminusnu + dmuWplusnu)/sm.sqrt(2), 
               (dmuWminusnu - dmuWplusnu)/sm.sqrt(2),  
               sm.sin(thetaW)*dmuAnu + sm.cos(thetaW)*dmuZnu])
dVmu, dVnu
Out[70]:
([2(βˆ‚ΞΌWΞ½++βˆ‚ΞΌWΞ½βˆ’)22(βˆ’βˆ‚ΞΌWΞ½++βˆ‚ΞΌWΞ½βˆ’)2βˆ‚ΞΌAΞ½sin⁑(ΞΈW)+βˆ‚ΞΌZΞ½cos⁑(ΞΈW)], [2(βˆ‚Ξ½WΞΌ++βˆ‚Ξ½WΞΌβˆ’)22(βˆ’βˆ‚Ξ½WΞΌ++βˆ‚Ξ½WΞΌβˆ’)2βˆ‚Ξ½AΞΌsin⁑(ΞΈW)+βˆ‚Ξ½ZΞΌcos⁑(ΞΈW)])
In [77]:
WW  = dVmu - dVnu - g2*Vmu.cross(Vnu)
WW
Out[77]:
[βˆ’g2(2(βˆ’WΞΌ++WΞΌβˆ’)(AΞ½sin⁑(ΞΈW)+ZΞ½cos⁑(ΞΈW))2βˆ’2(βˆ’WΞ½++WΞ½βˆ’)(AΞΌsin⁑(ΞΈW)+ZΞΌcos⁑(ΞΈW))2)+2(βˆ‚ΞΌWΞ½++βˆ‚ΞΌWΞ½βˆ’)2βˆ’2(βˆ‚Ξ½WΞΌ++βˆ‚Ξ½WΞΌβˆ’)2βˆ’g2(βˆ’2(WΞΌ++WΞΌβˆ’)(AΞ½sin⁑(ΞΈW)+ZΞ½cos⁑(ΞΈW))2+2(WΞ½++WΞ½βˆ’)(AΞΌsin⁑(ΞΈW)+ZΞΌcos⁑(ΞΈW))2)+2(βˆ’βˆ‚ΞΌWΞ½++βˆ‚ΞΌWΞ½βˆ’)2βˆ’2(βˆ’βˆ‚Ξ½WΞΌ++βˆ‚Ξ½WΞΌβˆ’)2βˆ‚ΞΌAΞ½sin⁑(ΞΈW)+βˆ‚ΞΌZΞ½cos⁑(ΞΈW)βˆ’βˆ‚Ξ½AΞΌsin⁑(ΞΈW)βˆ’βˆ‚Ξ½ZΞΌcos⁑(ΞΈW)βˆ’g2(βˆ’(βˆ’WΞΌ++WΞΌβˆ’)(WΞ½++WΞ½βˆ’)2+(WΞΌ++WΞΌβˆ’)(βˆ’WΞ½++WΞ½βˆ’)2)]
In [78]:
BB  = sm.cos(thetaW)*dmuAnu - sm.sin(thetaW)*dmuZnu
BB -= sm.cos(thetaW)*dnuAmu - sm.sin(thetaW)*dnuZmu
BB
Out[78]:
βˆ‚ΞΌAΞ½cos⁑(ΞΈW)βˆ’βˆ‚ΞΌZΞ½sin⁑(ΞΈW)βˆ’βˆ‚Ξ½AΞΌcos⁑(ΞΈW)+βˆ‚Ξ½ZΞΌsin⁑(ΞΈW)
In [80]:
VV = -sm.Rational(1, 4) * (BB*BB +  (WW.T*WW)[0])
VV = VV.expand()
VV = trigsimp(VV.subs({g2: q / sm.sin(thetaW)}))
VV
Out[80]:
βˆ’AΞΌ2(WΞ½+)2q24βˆ’AΞΌ2(WΞ½βˆ’)2q24+AΞΌAΞ½WΞΌ+WΞ½+q22+AΞΌAΞ½WΞΌβˆ’WΞ½βˆ’q22+AΞΌWΞΌ+WΞ½+ZΞ½q22tan⁑(ΞΈW)βˆ’AΞΌ(WΞ½+)2ZΞΌq22tan⁑(ΞΈW)+AΞΌWΞ½+βˆ‚ΞΌWΞ½βˆ’q2βˆ’AΞΌWΞ½+βˆ‚Ξ½WΞΌβˆ’q2+AΞΌWΞΌβˆ’WΞ½βˆ’ZΞ½q22tan⁑(ΞΈW)βˆ’AΞΌ(WΞ½βˆ’)2ZΞΌq22tan⁑(ΞΈW)βˆ’AΞΌWΞ½βˆ’βˆ‚ΞΌWΞ½+q2+AΞΌWΞ½βˆ’βˆ‚Ξ½WΞΌ+q2βˆ’AΞ½2(WΞΌ+)2q24βˆ’AΞ½2(WΞΌβˆ’)2q24βˆ’AΞ½(WΞΌ+)2ZΞ½q22tan⁑(ΞΈW)+AΞ½WΞΌ+WΞ½+ZΞΌq22tan⁑(ΞΈW)βˆ’AΞ½WΞΌ+βˆ‚ΞΌWΞ½βˆ’q2+AΞ½WΞΌ+βˆ‚Ξ½WΞΌβˆ’q2βˆ’AΞ½(WΞΌβˆ’)2ZΞ½q22tan⁑(ΞΈW)+AΞ½WΞΌβˆ’WΞ½βˆ’ZΞΌq22tan⁑(ΞΈW)+AΞ½WΞΌβˆ’βˆ‚ΞΌWΞ½+q2βˆ’AΞ½WΞΌβˆ’βˆ‚Ξ½WΞΌ+q2βˆ’(WΞΌ+)2(WΞ½βˆ’)2q24sin2⁑(ΞΈW)βˆ’(WΞΌ+)2ZΞ½2q24tan2⁑(ΞΈW)+WΞΌ+WΞ½+WΞΌβˆ’WΞ½βˆ’q22sin2⁑(ΞΈW)+WΞΌ+WΞ½+ZΞΌZΞ½q22tan2⁑(ΞΈW)+WΞΌ+WΞ½βˆ’βˆ‚ΞΌAΞ½q2+WΞΌ+WΞ½βˆ’βˆ‚ΞΌZΞ½q2tan⁑(ΞΈW)βˆ’WΞΌ+WΞ½βˆ’βˆ‚Ξ½AΞΌq2βˆ’WΞΌ+WΞ½βˆ’βˆ‚Ξ½ZΞΌq2tan⁑(ΞΈW)βˆ’WΞΌ+ZΞ½βˆ‚ΞΌWΞ½βˆ’q2tan⁑(ΞΈW)+WΞΌ+ZΞ½βˆ‚Ξ½WΞΌβˆ’q2tan⁑(ΞΈW)βˆ’(WΞ½+)2(WΞΌβˆ’)2q24sin2⁑(ΞΈW)βˆ’(WΞ½+)2ZΞΌ2q24tan2⁑(ΞΈW)βˆ’WΞ½+WΞΌβˆ’βˆ‚ΞΌAΞ½q2βˆ’WΞ½+WΞΌβˆ’βˆ‚ΞΌZΞ½q2tan⁑(ΞΈW)+WΞ½+WΞΌβˆ’βˆ‚Ξ½AΞΌq2+WΞ½+WΞΌβˆ’βˆ‚Ξ½ZΞΌq2tan⁑(ΞΈW)+WΞ½+ZΞΌβˆ‚ΞΌWΞ½βˆ’q2tan⁑(ΞΈW)βˆ’WΞ½+ZΞΌβˆ‚Ξ½WΞΌβˆ’q2tan⁑(ΞΈW)βˆ’(WΞΌβˆ’)2ZΞ½2q24tan2⁑(ΞΈW)+WΞΌβˆ’WΞ½βˆ’ZΞΌZΞ½q22tan2⁑(ΞΈW)+WΞΌβˆ’ZΞ½βˆ‚ΞΌWΞ½+q2tan⁑(ΞΈW)βˆ’WΞΌβˆ’ZΞ½βˆ‚Ξ½WΞΌ+q2tan⁑(ΞΈW)βˆ’(WΞ½βˆ’)2ZΞΌ2q24tan2⁑(ΞΈW)βˆ’WΞ½βˆ’ZΞΌβˆ‚ΞΌWΞ½+q2tan⁑(ΞΈW)+WΞ½βˆ’ZΞΌβˆ‚Ξ½WΞΌ+q2tan⁑(ΞΈW)βˆ’βˆ‚ΞΌAΞ½24+βˆ‚ΞΌAΞ½βˆ‚Ξ½AΞΌ2βˆ’(βˆ‚ΞΌWΞ½+)24+βˆ‚ΞΌWΞ½+βˆ‚Ξ½WΞΌ+2βˆ’(βˆ‚ΞΌWΞ½βˆ’)24+βˆ‚ΞΌWΞ½βˆ’βˆ‚Ξ½WΞΌβˆ’2βˆ’βˆ‚ΞΌZΞ½24+βˆ‚ΞΌZΞ½βˆ‚Ξ½ZΞΌ2βˆ’βˆ‚Ξ½AΞΌ24βˆ’(βˆ‚Ξ½WΞΌ+)24βˆ’(βˆ‚Ξ½WΞΌβˆ’)24βˆ’βˆ‚Ξ½ZΞΌ24

This is a complicated expression, but some conclusions are readily apparent such as the prediction of 3 and 4 vector boson interactions.

The QCD LagrangianΒΆ

The Lagrangian of quantum chromodynamics (QCD), for the 1st generation of particles, is given by

LQCD=UΒ―iΞ³ΞΌ(βˆ‚ΞΌ1+ig3Ξ»2β‹…GΞΌ)U+DΒ―iΞ³ΞΌ(βˆ‚ΞΌ1+ig3Ξ»2β‹…GΞΌ)Dβˆ’14GΞΌΞ½β‹…GΞΌΞ½and (GΞΌΞ½)a=(βˆ‚ΞΌGΞ½)aβˆ’(βˆ‚Ξ½GΞΌ)aβˆ’g3fabcGΞΌbGΞ½c,

where U and D, respectively, are the up and down quark field triplets each of whose components is associated with one of three quark colors, Ξ»a are the 8 Gell-Mann matrices each associated with one of the 8 gluon fields GΞΌa, and g3 is the strong coupling parameter. The quantities in parentheses are 3Γ—3 matrices. The QCD Lagrangian is invariant under SU(3) transformations and the numbers fabc are the structure constants of the group.

When QCD is added to the GSW (electroweak) Lagrangian, thereby forming the Lagrangian of the Standard Model (SM), the up and down quark fields acquire a 3-color index. Therefore, in the electroweak part of the Lagrangian each quark field will now carry a color index, in which case it is necessary to sum over the color indices in that part of the Lagrangian. Note that each quark field in the QCD Lagrangian is the sum of left-handed and right-handed fields.

Since the SM is gauge invariant, it is necessary to fix the definition of its bosonic (gauge) fields by choosing a gauge. Choosing a gauge is akin to choosing a coordinate system in that the predictions of the theory do not depend on the choice of gauge, at least in the limit of exact calculations. (In practice, calculations are done using perturbation theory. Consequently, depending on how the calculations are done there may be some residual, unphysical, gauge dependence in the predictions.) Unfortunately, choosing the gauge requires the addition of quite a bit of extra structure, involving unphysical fields called ghost fields, which makes what is otherwise an elegant structure much less so.

In [ ]:
 
In [ ]: